Gaussian unitary ensemble with two jump discontinuities, PDEs and the coupled Painlevé II and IV systems
aa r X i v : . [ m a t h - ph ] J un Gaussian unitary ensemble with two jump discontinuities,PDEs and the coupled Painlev´e II and IV systems
Shulin Lyu ∗ and Yang Chen † June 16, 2020
Abstract
We consider the Hankel determinant generated by the Gaussian weight with two jumpdiscontinuities. Utilizing the results of [C. Min and Y. Chen, Math. Meth. Appl. Sci. (2019), 301–321] where a second order PDE was deduced for the log derivative of the Hankeldeterminant by using the ladder operators adapted to orthogonal polynomials, we derive thecoupled Painlev´e IV system which was established in [X. Wu and S. Xu, arXiv: 2002.11240v2]by a study of the Riemann-Hilbert problem for orthogonal polynomials. Under double scaling,we show that, as n → ∞ , the log derivative of the Hankel determinant in the scaled variablestends to the Hamiltonian of a coupled Painlev´e II system and it satisfies a second order PDE. Inaddition, we obtain the asymptotics for the recurrence coefficients of orthogonal polynomials,which are connected with the solutions of the coupled Painlev´e II system. Keywords : Gaussian unitary ensembles; Hankel determinant; Painlev´e equations;Orthogonal polynomials
Mathematics Subject Classification 2020 : 33E17; 34M55; 42C05 ∗ School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai 519082, China; e-mail: [email protected] † Department of Mathematics, Faculty of Science and Technology, University of Macau, Macau, China; e-mail:[email protected] Introduction
The n -dimensional Gaussian unitary ensemble (GUE for short) is a set of n × n Hermitian randommatrices whose eigenvalues have the following joint probability density function p ( x , x , · · · , x n ) = 1 C n · n ! Y ≤ i 0. Through the Riemann-Hilbert (RH for short) formalismof orthogonal polynomials [13], they showed that σ n ( s , s ) + n ( s + s ) is the Hamiltonian of acoupled Painlev´e IV system. When s and s tend to the soft edge of the spectrum of GUE, byapplying Deift-Zhou nonlinear steepest descent analysis [11] to the RH problem (we call it RHmethod below), the asymptotic formulas for D n ( s , s ) and the associated orthogonal polynomialswere deduced, which are expressed in terms of the solution of a coupled Painlev´e II system.Comparing the finite n results of the above two papers concerning D n ( s , s ), we ask naturallywhether they are compatible with each other. To the best knowledge of the authors, it is not easyto obtain the second order PDE of [19] from the coupled Painlev´e IV system of [23]. What aboutthe other side? It transpires that the Hamiltonian of the coupled Painlev´e IV system of [23] canbe derived by using the results of [19]. This is the main purpose of the present paper, which mayprovide new insights into the connection between the ladder operator approach and RH problems.As we know, the ladder operator approach and the RH method are both very effective tools inthe study of unitary ensembles. The former is elementary in the sense that it uses the very basictheory of orthogonal polynomials and it provides a quite straightforward way to derive Painlev´etranscendents for finite dimensional problems particularly those involving one variable, for example,the gap probability of Gaussian and Jacobi unitary ensembles on ( − a, a ) with a > σ n ( s , s ) + n ( s + s ) is theHamiltonian of that system. Section 4 is devoted to the discussion of the double scaling limit of theHankel determinant. By using the finite n results given in section 2, we deduce that, as n → ∞ ,the log derivative of the Hankel determinant in the scaled variables tends to the Hamiltonian ofa coupled Painlev´e II system and it satisfies a second order PDE. In addition, for the recurrencecoefficients of the monic orthogonal polynomials associated with w ( x ; s , s ), we obtain their asymp-totic expansions in large n with the coefficients of the leading order term expressed in terms of thesolutions of the coupled Painlev´e II system. In this section, we present some results of [19] which will be used for our later derivation in subse-quent sections.Denote the Gaussian weight by w ( x ), i.e. w ( x ) := e − v ( x ) , v ( x ) = x . Then the weight function of our interest reads w ( x ; s , s ) = w ( x ) ( A + B θ ( x − s ) + B θ ( x − s )) . It is well known that the associated Hankel determinant admits the following representation (see[14, pp.16-19]) D n ( s , s ) = det (cid:18)Z ∞−∞ x i + j w ( x ; s , s ) dx (cid:19) n − i,j =0 = n − Y j =0 h j ( s , s ) . (2.1)Here h j ( s , s ) is the square of the L -norm of the j th-degree monic polynomial orthogonal withrespect to w ( x ; s , s ), namely, h j ( s , s ) δ jk := Z ∞−∞ P j ( x ; s , s ) P k ( x ; s , s ) w ( x ; s , s ) dx, (2.2)4or j, k = 0 , , , · · · , and P j ( x ; s , s ) := x j + p ( j, s , s ) x j − + · · · + P j (0; s , s ) . From the orthogonality, there follows the three term recurrence relation xP n ( x ; s , s ) = P n +1 ( x ; s , s ) + α n ( s , s ) P n ( x ; s , s ) + β n ( s , s ) P n − ( x ; s , s ) (2.3)with n ≥ 0, subject to the initial conditions P ( x ; s , s ) := 1 , β ( s , s ) P − ( x ; s , s ) := 0 . The recurrence coefficients are given by α n ( s , s ) = p ( n, s , s ) − p ( n + 1 , s , s ) , (2.4) β n ( s , s ) = h n ( s , s ) h n − ( s , s ) , (2.5)and it follows from (2.4) that n − X j =0 α j ( s , s ) = − p ( n, s , s ) . (2.6)For ease of notations, in the following discussion, we shall not display the s and s dependenceunless necessary.The recurrence relation implies the Christoffel-Darboux formula n − X j =0 P j ( x ) P j ( y ) h i = P n ( x ) P n − ( y ) − P n − ( x ) P n ( y ) h n − ( x − y ) . Here we point out that this identity and the recurrence relation hold for general monic polynomialsorthogonal with respect to any given positive function which has moments of all orders. See forexample [22, section 3.2] for more details.With all the above identities, one can derive a pair of ladder operators adapted to P n ( z ) = P n ( z ; s , s ): P ′ n ( z ) = β n A n ( z ) P n − ( z ) − B n ( z ) P n ( z ) ,P ′ n − ( z ) = ( B n ( z ) + v ′ ( z )) P n − ( z ) − A n − ( z ) P n ( z ) , ( z ) = z , A n ( z ) and B n ( z ) have simple poles at s and s , reading A n ( z ) = R n, ( s , s ) z − s + R n, ( s , s ) z − s + 2 ,B n ( z ) = r n, ( s , s ) z − s + r n, ( s , s ) z − s , with the residues defined by R n,i ( s , s ) := B i P n ( s i )e − s i h n , (2.7) r n,i ( s , s ) := B i P n ( s i ) P n − ( s i )e − s i h n − . (2.8)Here P j ( s i ) = P j ( x ; s , s ) | x = s i for j = n − , n . Moreover, one can show that A n ( z ) and B n ( z )satisfy three compatibility conditions( B n +1 ( z ) + B n ( z )) = ( z − α n ) A n ( z ) − v ′ ( z ) , ( S )1 + ( z − α n ) ( B n +1 ( z ) − B n ( z )) = β n +1 A n +1 ( z ) − β n A n − ( z ) , ( S ) B n ( z ) + v ′ ( z ) B n ( z ) + n − X j =0 A j ( z ) = β n A n ( z ) A n − ( z ) , ( S ′ )where ( S ′ ) results from ( S ) and ( S ). See [19, Theorem 3.1]. Concerning the discussion of ladderoperators and their compatibility conditions for general weight functions with jumps, refer to Lemma1, Remark 1 and Remark 2 of [2].Substituting A n ( z ) and B n ( z ) into ( S ) and ( S ′ ), by equating the residues on their both sides,it was found that the recurrence coefficients can be expressed in terms of the auxiliary quantitieswhich satisfy a system of difference equations (see [19], equations (3.7)-(3.14)). The results arepresented below. Proposition 2.1. (a) R n,i and r n,i , i = 1 , , satisfy the following system of difference equations: β n R n,i R n − ,i = r n,i , (2.9) r n +1 ,i + r n,i =( s i − α n ) R n,i . (2.10)(b) The recurrence coefficients are expressed in terms of R n,i and r n,i ( i = 1 , by α n = 12 ( R n, + R n, ) , (2.11) β n = 12 ( r n, + r n, + n ) . (2.12)6c) The quantity P n − j =0 ( R j, + R j, ) has the following representation n − X j =0 ( R j, + R j, ) = − s r n, − s r n, + 2 β n ( R n, + R n, + R n − , + R n − , ) . (2.13)By taking the derivatives of (2.2) with j = k = n and j = k + 1 = n , the auxiliary quantitiesturn out to be the partial derivatives of − ln h n ( s , s ) and ln p ( n, s , s ) with respect to s and s .Refer to equations (3.15), (3.16), (3.19), (3.20) of [19]. For ease of nations, in what follows, wedenote ∂∂s i and ∂ ∂s i ∂ sj ( i, j = 1 , 2) by ∂ s i and ∂ s i s j respectively. Proposition 2.2. The following differential relations hold ∂ s i ln h n ( s , s ) = − R n,i , (2.14) ∂ s i p ( n, s , s ) = r n,i , (2.15) with i = 1 , . In view of α n = p ( n, s , s ) − p ( n + 1 , s , s ) and β n = h n /h n − , it follows that ∂ s i α n ( s , s ) = r n,i − r n +1 ,i , (2.16) ∂ s i β n ( s , s ) = β n ( R n − ,i − R n,i ) . (2.17)Define σ n ( s , s ) := ( ∂ s + ∂ s ) ln D n ( s , s ) . With the fact that D n ( s , s ) = Q n − j =0 h j ( s , s ) and by using (2.11), one finds σ n ( s , s ) = − n − X j =0 ( R j, + R j, ) . (2.18)According to (2.6) and (2.14), there follows σ n ( s , s ) = 2 p ( n, s , s ) , (2.19)so that, in light of (2.15), ∂ s i σ n = 2 r n,i , i = 1 , . Hence, the compatibility condition ∂ s s σ n = ∂ s s σ n gives us ∂ s r n, = ∂ s r n, . (2.20)7ombining (2.13) with (2.18), and taking account of (2.9) and (2.12), we obtain the expression of σ n ( s , s ) in terms of the auxiliary quantities σ n = 2 (cid:18) s r n, + s r n, − r n, R n, − r n, R n, (cid:19) − ( r n, + r n, + n ) ( R n, + R n, ) . (2.21)By using the above identities, a second order PDE was established for σ n ( s , s ) (see Theorem3.3, [19]). Proposition 2.3. σ n ( s , s ) satisfies the following equation (cid:0) (2 s · ∂ s σ n + 2 s · ∂ s σ n − σ n ) − ∆ − ∆ (cid:1) = 4∆ ∆ , where ∆ and ∆ are defined by ∆ := (cid:0) ∂ s s σ n + ∂ s s σ n (cid:1) + 4 ( ∂ s σ n ) ( ∂ s σ n + ∂ s σ n + 2 n ) , ∆ := (cid:0) ∂ s s σ n + ∂ s s σ n (cid:1) + 4 ( ∂ s σ n ) ( ∂ s σ n + ∂ s σ n + 2 n ) . R n,i and Coupled Painlev´e IV system Based on the results presented in the previous section, we will derive a coupled PDEs satisfied by R n, and R n, in this section, which we will see in the next section are crucial for the derivation ofthe coupled Painlev´e II system under double scaling. We will also deduce the coupled Painlev´e IVsystem satisfied by quantities allied with R n,i and r n,i . R n,i and r n,i , and coupled PDEssatisfied by R n,i Combining the expressions involving the recurrence coefficients together, namely (2.11), (2.12),(2.16) and (2.17), with the aid of the difference equations (2.9) and (2.10), we arrive at the followingfour first order partial differential equations for R n,i and r n,i . Lemma 3.1. The quantities R n,i and r n,i , i = 1 , , satisfy the analogs of Riccati equations ∂ s i ( R n, + R n, ) =4 r n,i + ( R n, + R n, − s i ) R n,i , (3.1) ∂ s i ( r n, + r n, ) = 2 r n,i R n,i − ( n + r n, + r n, ) R n,i . (3.2)8 roof. Removing r n +1 ,i from (2.16) by using (2.10), we get ∂ s i α n ( s , s ) = 2 r n,i + ( α n − s i ) R n,i . Inserting (2.11) into the above equation, we obtain (3.1).Getting rid of R n − ,i in (2.17) by using (2.9), we find ∂ s i β n = r n,i R n,i − β n R n,i . Plugging (2.12) into this identity, we come to (3.2).From (3.1), we readily get the expressions of r n,i in terms of R n,i and their first order partialderivatives. Substituting them into (3.2), we arrive at a coupled PDEs satisfied by R n,i . Theorem 3.2. The quantities R n,i , i = 1 , , satisfy the following coupled PDEs: (cid:0) ∂ s s + ∂ s s (cid:1) ( R n, + R n, ) − ∂ s ( R n, + R n, ) · (cid:18) ∂ s ( R n, + R n, )2 R n, + R n, (cid:19) + 2( s − s ) ( ∂ s R n, )+ R n, (cid:18) ∂ s ( R n, + R n, ) − 32 ( R n, + R n, ) + 2 (cid:0) s R n, + ( s + s ) R n, − s + 2 n + 1 (cid:1)(cid:19) = 0 , (3.3a) and (cid:0) ∂ s s + ∂ s s (cid:1) ( R n, + R n, ) − ∂ s ( R n, + R n, ) · (cid:18) ∂ s ( R n, + R n, )2 R n, + R n, (cid:19) + 2( s − s ) ( ∂ s R n, )+ R n, (cid:18) ∂ s ( R n, + R n, ) − 32 ( R n, + R n, ) + 2 (cid:0) ( s + s ) R n, + 2 s R n, − s + 2 n + 1 (cid:1)(cid:19) = 0 . (3.3b) Remark 1. Interchanging s with s , R n, with R n, in (3.3a) , we get (3.3b) . This observationagrees with the symmetry in position of s and s in the weight function w ( x ; s , s ) and the defini-tions of R n,i . Remark 2. If B = 0 , then R n, = 0 and R n, depends only on s . Equation (3.3a) is reduced toan ordinary differential equation satisfied by R n ( s ) := R n, ( s , R ′′ n = ( R ′ n ) R n + 32 R n − s R n + 2( s − n − R n , (3.4) which is identical with (2.37) of [19] where t is used instead of s . As was pointed out there, (3.4) can be transformed into a Painlev´e IV equation satisfied by y ( s ) := R n ( − s ) .In case B = 0 , via a similar argument, we find that R n, (0 , s ) satisfies (3.4) with s replacedby s . .2 Coupled Painlev´e IV system Define x := s + s , s := s − s , and introduce four quantities allied with R n,i ( s , s ) and r n,i ( s , s ): a i ( x, s ) := r n,i R n,i ( r n, + r n, + n ) ,b i ( x, s ) := R n,i r n,i ( r n, + r n, + n ) , with i = 1 , 2. We have s = x − s, s = x + s, and R n,i ( s , s ) = a i b i a b + a b + n ,r n,i ( s , s ) = a i b i . (3.5)By making use of the results from section 2, we show that a i and b i satisfy a coupled Painlev´eIV system with ( ∂ s + ∂ s ) ln D n ( s , s ) + n ( s + s ) being the Hamiltonian. Theorem 3.3. The quantity H IV ( a , a , b , b ; x, s ) := σ n ( s , s ) + n ( s + s ) with σ n ( s , s ) = ( ∂ s + ∂ s ) ln D n ( s , s ) satisfying the second order PDE given by Proposition 2.3,is expressed in terms of a i ( x, s ) and b i ( x, s ) by H IV ( a , a , b , b ; x, s ) = − a b + a b + n )( a + a ) − ( a b + a b )+ 2 (( x − s ) a b + ( x + s ) a b + nx ) , (3.6) and it is the Hamiltonian of the following coupled Painlev´e IV system ∂ x a = ∂ b H IV = − a ( a + a + b − x + s ) ,∂ x a = ∂ b H IV = − a ( a + a + b − x − s ) ,∂ x b = ∂ a H IV = b + 2 b (2 a + a − x + s ) + 2( a b + n ) ,∂ x b = ∂ a H IV = b + 2 b ( a + 2 a − x − s ) + 2( a b + n ) . (3.7)10xpression (3.6) follows directly from (2.21) and (3.5). To derive the coupled Painlev´e IV system,we shall establish four linear equations in the variables ∂ x a i and ∂ x b i , i = 1 , 2. Before proceedingfurther, we first present some results which will be used later for the derivation.Since ∂ x = ∂ s + ∂ s , we readily get from (2.14) and (2.15) that ∂ x ln h n ( s , s ) = − ( R n, + R n, ) , (3.8) ∂ x p ( n, s , s ) = r n, + r n, . (3.9)Noting that α n = p ( n, s , s ) − p ( n + 1 , s , s ) and β n = h n /h n − , we find ∂ x α n ( s , s ) = X i =1 , ( r n,i − r n +1 ,i ) , (3.10) ∂ x β n ( s , s ) = β n X i =1 , ( R n − ,i − R n,i ) . (3.11)As an immediate consequence of (2.11) and (2.12), we have Lemma 3.4. The recurrence coefficients are expressed in terms of a i and b i by α n = a b + a b a b + a b + n ) , (3.12) β n = 12 ( a b + a b + n ) . (3.13)Replacing r n, + r n, + n by 2 β n in the definitions of a i ( x, s ), which is due to (2.12), with theaid of (2.9), we build the direct relationships between a i and the quantities with index n − 1, i.e. R n − ,i , α n − and ∂ x ln h n − . . Lemma 3.5. We have a i ( x, s ) = R n − ,i , i = 1 , , (3.14) so that, in view of (2.11) and (3.8) , α n − ( s , s ) = a ( x, s ) + a ( x, s ) = − ∂ x h n − ( s , s ) . (3.15)Now we are ready to deduce the four linear equations in ∂ x a i and ∂ x b i , each of which will bestated as a lemma. We start from the combination of (3.12) and (3.13) which gives us a b + a b = 4 α n β n . (3.16)11 emma 3.6. We have b ( ∂ x a ) + b ( ∂ x a ) + 2 a b ( ∂ x b ) + 2 a b ( ∂ x b )=4( a a + b b + n )( a a + b b ) + 2 a b ( a + a − s ) + 2 a b ( a + a − s ) . Proof. Taking the derivative on both sides of (3.16) with respect to x , we have ∂ x (cid:0) a b + a b (cid:1) = 4 β n ( ∂ x α n ) + 4 α n ( ∂ x β n ) . (3.17)Now we shall make use of (3.10) and (3.11) to derive the expressions of ∂ x α n and ∂ x β n in terms of a i , b i or R n,i , r n,i . Using (2.10) to get rid of r n +1 ,i , i = 1 , , in (3.10), we find ∂ x α n ( s , s ) = X i =1 , (2 r n,i + ( α n − s i ) R n,i ) . On account of (3.14), we replace R n − ,i by 2 a i in (3.11) and get ∂ x β n ( s , s ) = − β n ( R n, + R n, ) + 2 β n ( a + a ) . Plugging the above two identities into (3.17), we obtain b ( ∂ x a ) +2 a b ( ∂ x b ) + b ( ∂ x a ) + 2 a b ( ∂ x b )=4 β n (2( r n, + r n, ) − s R n, − s R n, ) + 8 α n β n ( a + a ) . On substituting (3.5), (3.12) and (3.13) into this equation, we come to the desired result.Replacing n by n − ∂ x α n − ( s , s ) = X i =1 , ( r n − ,i − r n,i ) ,r n,i + r n − ,i = ( s i − α n − ) R n − ,i . Using the second equality to remove r n − ,i in the first one, we are led to ∂ x α n − ( s , s ) = X i =1 , (( s i − α n − ) R n − ,i − r n,i ) . According to (3.15) and (3.14), we replace α n − by a + a and R n − ,i by 2 a i in the above identity.By taking note that r n,i = a i b i , i = 1 , 2, we come to the following equation.12 emma 3.7. We have ∂ x a + ∂ x a = − a ( a + a + b − s ) − a ( a + a + b − s ) . The next equation is obtained by combining the two expressions involving β n and ∂ x β n . Lemma 3.8. We have b i ( ∂ x a i ) + a i ( ∂ x b i ) =2 a i ( a b + a b + n ) − a i b i , i = 1 , . Proof. Plugging (2.12) into (2.17), we get ∂ s i ( r n, + r n, ) = ( r n, + r n, + n ) ( R n − ,i − R n,i ) , i = 1 , . In view of (2.20), i.e. ∂ s r n, = ∂ s r n, , we find( ∂ s + ∂ s ) r n, = ( r n, + r n, + n ) ( R n − , − R n, ) , ( ∂ s + ∂ s ) r n, = ( r n, + r n, + n ) ( R n − , − R n, ) . Since ∂ x = ∂ s + ∂ s , it follows that ∂ x r n,i = ( r n, + r n, + n ) ( R n − ,i − R n,i ) , i = 1 , . Using (3.5) to replace r n,i and R n,i in this expression, and substituting 2 a i for R n − ,i , which is dueto (3.14), we complete the proof. Proof of Theorem 3.3 Now we have four linear equations in ∂ x a , ∂ x a , ∂ x b and ∂ x b , namely, b ( ∂ x a ) + b ( ∂ x a ) +2 a b ( ∂ x b ) + 2 a b ( ∂ x b )=4( a a + b b + n )( a a + b b ) + 2 a b ( a + a − s ) + 2 a b ( a + a − s ) , (3.18) ∂ x a + ∂ x a = − a ( a + a + b − s ) − a ( a + a + b − s ) , (3.19) b ( ∂ x a ) + a ( ∂ x b ) =2 a ( a b + a b + n ) − a b , (3.20) b ( ∂ x a ) + a ( ∂ x b ) =2 a ( a b + a b + n ) − a b . (3.21)Subtracting (3.18) from the sum of (3.20) multiplied by 2 b and (3.21) multiplied by 2 b , we get b ( ∂ x a ) + b ( ∂ x a ) = − a b ( a + a + b − s ) − a b ( a + a + b − s ) . (3.22)Combining (3.19) with (3.22) to solve for ∂ x a and ∂ x a , and substituting the resulting expressionsinto (3.20) and (3.21), we arrive at the desired coupled Painlev´e IV system (3.7). (cid:3) emark 3. The Hamiltonian of the coupled Painlev´e IV system presented in Theorem 3.3 is thesame as the one given by (1.15) and (1.16) of [23] which was derived via the Riemann-Hilbertapproach. Taking note that our symbols β n and h n correspond to β n and γ − n of [23], we findthat our equations (3.12) , (3.13) and (3.15) are consistent with (1.23), (1.24) and (1.26) of [23]respectively. We remind the reader that our weight function is obtained by multiplying the Gaussian weight bya factor with two jumps, i.e. w ( x ; s , s ) = e − x ( A + B θ ( x − s ) + B θ ( x − s )) , where B B = 0. In this section, we discuss the asymptotic behavior of the associated Hankeldeterminant when s and s tend to the soft edge of the spectrum of GUE, namely, s i := √ n + t i √ n / , i = 1 , . This double scaling may be explained in the following way. As we know, the classical Hermitepolynomials H n ( x ) are orthogonal with respect to the Gaussian weight e − x , x ∈ ( −∞ , ∞ ). Underthe double scaling x = √ n + t √ n / and as n → ∞ , the Hermite function e − x / H n ( x ) is approx-imated by the Airy function A ( x ) multiplied by a factor involving n [22, Formula (8.22.14)]. Seealso [12, formula (3.6)] and [21, Theorem 2.1] for more explanation about this double scaling.When B = 0 or B = 0, our weight function has only one jump. This case was studied in [19]and the expansion formula for R n ( s ) := R n, ( s , 0) in large n was given by R n ( s ) = n − / v ( t ) + n − / v ( t ) + n − / v ( t ) + O (cid:0) n − / (cid:1) . It was obtained by using the second order ordinary differential equation satisfied by R n ( s ) (seeTheorem 2.10, [19]). Hence, for our two jump case where B B = 0, we assume R n, ( s , s ) = ∞ X i =1 µ i ( t , t ) · n (1 − i ) / , (4.1a) R n, ( s , s ) = ∞ X i =1 ν i ( t , t ) · n (1 − i ) / . (4.1b)14rom the compatibility condition ∂ s s R n,i = ∂ s s R n,i , i = 1 , 2, it follows that ∂ t t µ ( t , t ) = ∂ t t µ ( t , t ) ,∂ t t ν ( t , t ) = ∂ t t ν ( t , t ) . (4.2)We keep these two relations in mind in the subsequent discussions.Substituting (4.1) into the left hand side of (3.3a) and (3.3b), by taking their series expansionsin large n and setting the leading coefficients to be zero, we get a coupled PDEs satisfied by µ and ν . Theorem 4.1. The leading coefficients in the expansions of R n,i in large n , i.e. µ ( t , t ) = lim n →∞ n / R n, ( s , s ) ,ν ( t , t ) = lim n →∞ n / R n, ( s , s ) , satisfy the following coupled PDEs (cid:0) ∂ t t + ∂ t t (cid:1) ( µ + ν ) − ( ∂ t ( µ + ν )) µ + 2 µ ( √ µ + ν ) − t ) = 0 , (4.3a) (cid:0) ∂ t t + ∂ t t (cid:1) ( µ + ν ) − ( ∂ t ( µ + ν )) ν + 2 ν ( √ µ + ν ) − t ) = 0 . (4.3b)Plugging (4.1) into (3.1), we get r n, ( s , s ) = µ √ n / + µ √ √ ∂ t ( µ + ν ) + O ( n − / ) , (4.4a) r n, ( s , s ) = ν √ n / + ν √ √ ∂ t ( µ + ν ) + O ( n − / ) . (4.4b)Hence, according to (2.20), i.e. ∂ s r n, = ∂ s r n, , we find ∂ t µ ( t , t ) = ∂ t ν ( t , t ) , (4.5)To continue, we define v ( t , t − t ) := − µ ( t , t ) √ ,v ( t , t − t ) := − ν ( t , t ) √ . With the aid of (4.5), we establish the following differential relations.15 emma 4.2. We have ∂ t ( µ ( t , t ) + ν ( t , t )) = − √ v ξ ( t , t − t ) , (4.6a) ∂ t ( µ ( t , t ) + ν ( t , t )) = − √ v ξ ( t , t − t ) , (4.6b) where v iξ ( i = 1 , denotes the first order derivative of v i ( ξ, η ) with respect to ξ .Proof. By the definition of v ( t , t − t ), we find ∂ t µ ( t , t ) = − √ · ∂ t v ( t , t − t )= − √ v ξ ( t , t − t ) − v η ( t , t − t )) ,∂ t µ ( t , t ) = − √ · ∂ t v ( t , t − t )= − √ · v η ( t , t − t ) , so that ( ∂ t + ∂ t ) µ ( t , t ) = −√ v ξ ( t , t − t ) . In view of (4.5), we obtain (4.6a). Via a similar argument, we can prove (4.6b).With the aid of (4.5) and (4.6), we establish the following equations for v and v by using thecoupled PDEs (4.3). Theorem 4.3. The quantities v ( t , t − t ) and v ( t , t − t ) satisfy a coupled nonlinear equations v iξξ − v iξ v i − v i (2( v + v ) + t i ) = 0 , (4.7) where v iξ and v iξξ denote the first and second order derivative of v i ( ξ, η ) with respect to ξ respectively.Proof. Differentiation of both sides of (4.6a) over t and t gives us ∂ t t ( µ ( t , t ) + ν ( t , t )) = − √ v ξξ ( t , t − t ) − v ξη ( t , t − t )) ,∂ t t ( µ ( t , t ) + ν ( t , t )) = − √ v ξη ( t , t − t ) , where in the second equality we make use of (4.2). It follows that (cid:0) ∂ t t + ∂ t t (cid:1) ( µ ( t , t ) + ν ( t , t )) = −√ v ξξ ( t , t − t ) . (4.8a)16imilarly, by differentiating both sides of (4.6b) over t and t , we get (cid:0) ∂ t t + ∂ t t (cid:1) ( µ ( t , t ) + ν ( t , t )) = −√ v ξξ ( t , t − t ) . (4.8b)Plugging (4.8) and (4.6) into (4.3), we arrive at the desired equations.Now we look at σ n ( s , s ) which is defined by σ n ( s , s ) := ( ∂ s + ∂ s ) ln D n ( s , s ) . Recall that it is expressed in terms of R n,i and r n,i by (2.21). Substituting the expansions of R n,i and r n,i into this expression, we establish the following results. Theorem 4.4. σ n ( s , s ) has the following asymptotic expansion in large nσ n ( s , s ) = √ n / H II ( t , t − t ) + O ( n − / ) , (4.9) where H II ( t , t − t ) is the Hamiltonian of the following coupled Painlev´e II system v iξ = ∂H II ∂w i = 2 v i w i , (4.10a) w iξ = − ∂H II ∂v i = 2 ( v + v ) + t i − w i , (4.10b) which is given by H II ( t , t − t ) = v w + v w − ( v + v ) − t v − t v . (4.11) Here v i = v i ( t , t − t ) and w i = w i ( t , t − t ) . Moreover, H II satisfies the following second ordersecond degree PDE ( ∂ t H II ) · (cid:0) ∂ t t H II + ∂ t t H II (cid:1) + ( ∂ t H II ) · (cid:0) ∂ t t H II + ∂ t t H II (cid:1) = 4 ( ∂ t H II ) ( ∂ t H II ) ( t · ∂ t H II + t · ∂ t H II − H II ) . (4.12) Proof. Recall (2.21), i.e. σ n ( s , s ) = 2 (cid:18) s r n, + s r n, − r n, R n, − r n, R n, (cid:19) − ( r n, + r n, + n ) ( R n, + R n, ) . Substituting (4.1) and (4.4) into the right hand side of this expression, by taking its series expansionin large n , we obtain σ n ( s , s ) = − ( ∂ t ( µ + ν )) µ − ( ∂ t ( µ + ν )) ν − ( µ + ν ) √ t µ + t ν ! n / + O (cid:0) n − / (cid:1) . n / by using (4.6), and substituting −√ v and −√ v for µ and ν respectively, we find σ n ( s , s ) = √ n / (cid:18) v ξ v + v ξ v − ( v + v ) − t v − t v (cid:19) + O (cid:0) n − / (cid:1) . On writing w i ( t , t − t ) := v iξ ( t , t − t )2 v i ( t , t − t ) , we get (4.9).From the above definition of w i , we readily see that (4.10a) holds. Taking the derivative on bothsides of (4.10a), we are led to v iξξ = 4 v i w i + 2 v i w iξ . Inserting it and (4.10a) into (4.7), after simplification, we produce (4.10b).To derive (4.12), we plugging (4.9) into the PDE satisfied by σ n , i.e. (2.3). By taking the seriesexpansion of its left hand side and setting the leading coefficient to be zero, we obtain (4.12).Recall (2.11) and (2.12) which express the recurrence coefficients in terms of R n,i and r n,i ,namely, α n = 12 ( R n, + R n, ) ,β n = 12 ( r n, + r n, + n ) . Substituting (4.1) and (4.4) into the above expressions, after simplification, we get the asymptoticexpansions of α n and β n in large n . Theorem 4.5. The recurrence coefficients of the monic polynomials orthogonal with respect to theGaussian weight with two jump discontinuities have the following asymptotics for large nα n ( s , s ) = − v + v √ n / + O ( n − / ) ,β n ( s , s ) = n − v + v n / + O (1) . 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